No side have made more than the 344 for one posted by the West Indies against England in 1984 to win in the fourth innings of a Test at Lord's.
New Zealand bowled England out for 478 in their second innings on Monday after the hosts resumed on 429 for six.
Left-arm paceman Trent Boult took all the wickets to fall during a spell of four wickets for nine runs in 17 balls.
Boult finished the innings with fine figures of five for 85 in 34 overs.
England captain Alastair Cook was among his wickets on Monday after the left-handed opener extended his overnight 153 not out to 162 before he was caught behind.
This is the first of a two-Test series. – AFP
Original source: Record chase awaits Black Caps